3.145 \(\int \frac{x^{-m}}{a^5+x^5} \, dx\)

Optimal. Leaf size=46 \[ \frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{5},\frac{6-m}{5},-\frac{x^5}{a^5}\right )}{a^5 (1-m)} \]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/5, (6 - m)/5, -(x^5/a^5)])/(a^5*(1 - m))

_______________________________________________________________________________________

Rubi [A]  time = 0.0280382, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{x^{1-m} \, _2F_1\left (1,\frac{1-m}{5};\frac{6-m}{5};-\frac{x^5}{a^5}\right )}{a^5 (1-m)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^m*(a^5 + x^5)),x]

[Out]

(x^(1 - m)*Hypergeometric2F1[1, (1 - m)/5, (6 - m)/5, -(x^5/a^5)])/(a^5*(1 - m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.13645, size = 31, normalized size = 0.67 \[ \frac{x^{- m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{m}{5} + \frac{1}{5} \\ - \frac{m}{5} + \frac{6}{5} \end{matrix}\middle |{- \frac{x^{5}}{a^{5}}} \right )}}{a^{5} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**m)/(a**5+x**5),x)

[Out]

x**(-m + 1)*hyper((1, -m/5 + 1/5), (-m/5 + 6/5,), -x**5/a**5)/(a**5*(-m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0262242, size = 47, normalized size = 1.02 \[ -\frac{x^{1-m} \text{Hypergeometric2F1}\left (1,\frac{1-m}{5},\frac{1-m}{5}+1,-\frac{x^5}{a^5}\right )}{a^5 (m-1)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^m*(a^5 + x^5)),x]

[Out]

-((x^(1 - m)*Hypergeometric2F1[1, (1 - m)/5, 1 + (1 - m)/5, -(x^5/a^5)])/(a^5*(-
1 + m)))

_______________________________________________________________________________________

Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{m} \left ({a}^{5}+{x}^{5} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^m)/(a^5+x^5),x)

[Out]

int(1/(x^m)/(a^5+x^5),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-m}}{a^{5} + x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^5 + x^5)*x^m),x, algorithm="maxima")

[Out]

integrate(x^(-m)/(a^5 + x^5), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a^{5} + x^{5}\right )} x^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^5 + x^5)*x^m),x, algorithm="fricas")

[Out]

integral(1/((a^5 + x^5)*x^m), x)

_______________________________________________________________________________________

Sympy [A]  time = 68.0489, size = 92, normalized size = 2. \[ - \frac{m x x^{- m} \Phi \left (\frac{x^{5} e^{i \pi }}{a^{5}}, 1, - \frac{m}{5} + \frac{1}{5}\right ) \Gamma \left (- \frac{m}{5} + \frac{1}{5}\right )}{25 a^{5} \Gamma \left (- \frac{m}{5} + \frac{6}{5}\right )} + \frac{x x^{- m} \Phi \left (\frac{x^{5} e^{i \pi }}{a^{5}}, 1, - \frac{m}{5} + \frac{1}{5}\right ) \Gamma \left (- \frac{m}{5} + \frac{1}{5}\right )}{25 a^{5} \Gamma \left (- \frac{m}{5} + \frac{6}{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**m)/(a**5+x**5),x)

[Out]

-m*x*x**(-m)*lerchphi(x**5*exp_polar(I*pi)/a**5, 1, -m/5 + 1/5)*gamma(-m/5 + 1/5
)/(25*a**5*gamma(-m/5 + 6/5)) + x*x**(-m)*lerchphi(x**5*exp_polar(I*pi)/a**5, 1,
-m/5 + 1/5)*gamma(-m/5 + 1/5)/(25*a**5*gamma(-m/5 + 6/5))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a^{5} + x^{5}\right )} x^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^5 + x^5)*x^m),x, algorithm="giac")

[Out]

integrate(1/((a^5 + x^5)*x^m), x)